cousinhaui
2022-10-21
Answered

Let g be a probability density function, what is the ratio $\frac{g(x)}{g(t)}$ for $x>t$?

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Jovanni Salinas

Answered 2022-10-22
Author has **18** answers

Step 1

Continuity is important because we can always take a nice-behaved pdf like the Gaussian, and adjust it over a set of points with zero measure to get a PDF that doesn't strictly have a tail limit.

For example. Let $\varphi (x)$be the standard normal density function. It has nice tail properties and adheres to your conjecture.

Now, lets form the following function

$$Q(x)=x{\mathbf{1}}_{x\in \mathbb{Z}}$$

Step 2

Then the function $\psi (x)+Q(x)$ has not limit at either tail, yet it integrates to 1 and all that good stuff, since the domian where $Q(x)>0$ has zero Lebesgue measure.

So if g is a continuous pdf then limits should exist.

Continuity is important because we can always take a nice-behaved pdf like the Gaussian, and adjust it over a set of points with zero measure to get a PDF that doesn't strictly have a tail limit.

For example. Let $\varphi (x)$be the standard normal density function. It has nice tail properties and adheres to your conjecture.

Now, lets form the following function

$$Q(x)=x{\mathbf{1}}_{x\in \mathbb{Z}}$$

Step 2

Then the function $\psi (x)+Q(x)$ has not limit at either tail, yet it integrates to 1 and all that good stuff, since the domian where $Q(x)>0$ has zero Lebesgue measure.

So if g is a continuous pdf then limits should exist.

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